**2. Mathematical Modeling of Carbon Flux**

Carbon flux is the main parameter that influences model calculations and determines the control precision of the carburizing process. The transfer process of carbon atoms from the atmosphere to the workpiece surface can be defined by three types of boundary conditions. In typical atmospheric carburizing, the carbon potential control method is used, i.e., the third type of boundary condition. The difference between the carbon concentration of the atmosphere and the carbon concentration of the metal surface acts as the driving force for carbon to enter into the steel from the gas phase. During carburization, an oxygen probe is used to measure the carbon concentration in the atmosphere, and the steel foil method is applied to reverse calculate the transfer coefficient, thus achieving the detection and control of the carburization process [8].

Owing to the vacuum environment of low-pressure vacuum carburizing, the carburizing gas concentration is low, and the carbon concentration in the atmosphere cannot be measured using an oxygen probe. In this scenario, the second type of boundary condition is used, i.e., the carbon flux. Different from atmospheric carburizing, low-pressure vacuum carburizing has a low carburizing pressure. The number of carburizing gas molecules is extremely small and the molecules can be quickly decomposed and absorbed onto the steel surface, instantly boosting the surface carbon flux to the maximum. Therefore, the surface carbon flux model normally used for atmospheric carburizing is no longer applicable [22]. The value of carbon flux is related to the carburizing gas type, pressure, temperature, alloy type, and surface state. To study low-pressure vacuum carburizing, the overall average carbon flux method is generally used [23], as shown in Equation (1). This method obtains the carbon flux by experimentally measuring the mass increment before and after a certain period of carburization.

$$J = \frac{\Delta m}{\mathcal{S} \cdot t}. \tag{1}$$
